Fixed Point Theorems for Plane Continua with Applications

Blokh, Alexander; Fokkink, Robbert; Mayer, John C.; Oversteegen, Lex G.; Tymchatyn, E. D.

doi:10.1090/s0065-9266-2012-00671-x

Cited by 28 publications

(49 citation statements)

References 36 publications

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“…Since positively oriented maps generalize orientation preserving homeomorphisms their results are related to ours, but the emphasis of their approach is more on the structure of X, expressed in one-step "'geometric"' conditions, whereas our approach can be considered more of dynamical nature, as determined by iterations of the homeomorphism on X. The reader is referred to [8] for more details.…”

Section: Introductionmentioning

confidence: 91%

“…Note that Theorem A holds true for orientation reversing homeomorphism, by the result of H. Bell [5], and K. Kuperberg generalized Bell's result to plane separating continua [18], [19]. However, it remains a major 100-year-old open problem if every acyclic planar continuum has the fixed point property (see Scottish Book Problem 107 [20], and [8] for the most recent developments regarding the subject).…”

Section: Introductionmentioning

confidence: 98%

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On a generalization of the Cartwright–Littlewood fixed point theorem for planar homeomorphisms

Boroński

2016

Ergod. Th. Dynam. Sys.

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Abstract. We prove a generalization of the fixed point theorem of Cartwright and Littlewood. Namely, suppose that h : R 2 → R 2 is an orientation preserving planar homeomorphism, and let C be a continuum such that h −1 (C) ∪ C is acyclic. If there is a c ∈ C such that {h −i (c) : i ∈ N} ⊆ C, or {h i (c) : i ∈ N} ⊆ C, then C also contains a fixed point of h. Our approach is based on Brown's short proof of the result of Cartwright and Littlewood. In addition, making use of a linked periodic orbits theorem of Bonino, we also prove a counterpart of the aforementioned result for orientation reversing homeomorphisms, that guarantees a 2-periodic orbit in C if it contains a k-periodic orbit (k > 1).

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Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 98%

On a generalization of the Cartwright–Littlewood fixed point theorem for planar homeomorphisms

Boroński

2016

Ergod. Th. Dynam. Sys.

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“…In this section, we prove that bounded components of F λ \ P λ of the first three types do not exist. We use [BFMOT12], where fixed and periodic points of various maps of plane continua were studied. Definition 4.2.…”

Section: Bounded Components Of F λ \ P λ Must Be Of Siegel Capture Tymentioning

confidence: 99%

Complementary components to the cubic principal hyperbolic domain

Blokh

Oversteegen

Ptacek

et al. 2018

Proc. Amer. Math. Soc.

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We study the closure of the cubic Principal Hyperbolic Domain and its intersection P λ with the slice F λ of the space of all cubic polynomials with fixed point 0 defined by the multiplier λ at 0. We show that any bounded domain W of F λ \ P λ consists of J-stable polynomials f with connected Julia sets J(f ) and is either of Siegel capture type (then f ∈ W has an invariant Siegel domain U around 0 and another Fatou domain V such that f | V is twoto-one and f k (V ) = U for some k > 0) or of queer type (then a specially chosen critical point of f ∈ W belongs to J(f ), the set J(f ) has positive Lebesgue measure, and carries an invariant line field). 1 2 A. BLOKH, L. OVERSTEEGEN, R. PTACEK, AND V. TIMORINhas a non-repelling fixed point. As follows from the Douady-Hubbard parameter landing theorem [DH8485, Hub93], the Mandelbrot set itself can be thought of as the union of the main cardioid and limbs (connected components of M 2 \ PHD 2 ) parameterized by reduced rational fractions p/q ∈ (0, 1).This motivates our study of higher degree analogs of PHD 2 started in [BOPT14]. More precisely, complex numbers c are in one-to-one correspondence with affine conjugacy classes of quadratic polynomials (throughout we call affine conjugacy classes of polynomials classes of polynomials). Thus a natural higher-degree analog of the set M 2 is the degree d connectedness locus M d defined as the set of classes of degree d polynomials f , all of whose critical points do not escape, or, equivalently, whose Julia set J(f ) is connected. The Principal Hyperbolic Domain PHD d of M d is defined as the set of classes of hyperbolic degree d polynomials with Jordan curve Julia sets. Equivalently, the class [f ] of a degree d polynomial f belongs to PHD d if all critical points of f are in the immediate attracting basin of the same attracting (or super-attracting) fixed point. In [BOPT14] we describe properties of cubic polynomials f such that [f ] ∈ PHD d ; notice that Theorem 1.1 holds for any d 2.Theorem 1.1 ([BOPT14]). If [f ] ∈ PHD d , then f has a non-repelling fixed point, no repelling periodic cutpoints in J(f ), and all its non-repelling periodic points, except at most one fixed point, have multiplier 1.

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“…We will show how to modify some of the results of [BFMOT10] to our needs. However first we need a few definitions introduced in [BFMOT10].…”

Section: Definition 22 (Dynamics and Invariant Laminations) A Laminmentioning

confidence: 99%

“…(1) Suppose that f maps a dendrite Yet another result from [BFMOT10] is Lemma 7.2.2(1) which is stated below. When talking about points in a dendrite D, we say that a point x separates a point y from a point z if y and z belong to distinct components of D \ {x}.…”

Section: Definition 24 (Boundary Scrambling For Dendrites) Suppose mentioning

confidence: 99%